Kite's ups and downs notation: Difference between revisions
Wikispaces>TallKite **Imported revision 558113783 - Original comment: ** |
Wikispaces>TallKite **Imported revision 558118067 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2015-09-03 | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2015-09-03 05:58:30 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>558118067</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></tt><br> | ||
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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The advantage to this notation is that you always know where your fifth is. And hence your 4th, and your major 9th, hence the maj 2nd and the min 7th too. You have convenient landmarks to find your way around, built into the notation. The notation is a map of unfamiliar territory, and we want this map to be as easy to read as possible. | The advantage to this notation is that you always know where your fifth is. And hence your 4th, and your major 9th, hence the maj 2nd and the min 7th too. You have convenient landmarks to find your way around, built into the notation. The notation is a map of unfamiliar territory, and we want this map to be as easy to read as possible. | ||
The basic pattern for 22-EDO is P1-m2-^m2-vM2-M2-m3-^m3-vM3-M3-P4-d5-^d5-vP5-P5 etc. That's pronounced "upminor 2nd, downmajor 3rd", etc. The ups and downs are leading in relative notation but trailing in absolute notation. You can apply this pattern to any key, with certain keys requiring double-sharps or even triple-sharps. The mid notes always form a | The basic pattern for 22-EDO is P1-m2-^m2-vM2-M2-m3-^m3-vM3-M3-P4-d5-^d5-vP5-P5 etc. That's pronounced "upminor 2nd, downmajor 3rd", etc. The ups and downs are leading in relative notation but trailing in absolute notation. You can apply this pattern to any key, with certain keys requiring double-sharps or even triple-sharps. The mid notes always form a chain of fifths. | ||
You can loosely relate the ups and downs to JI: major = red or fifthward white, downmajor = yellow, upminor = green, minor = blue or fourthwards white. Or simply up = green, down = yellow, and mid = white, blue or red. (See [[Kite's color notation]] for an explanation of the colors.) These correlations are for 22-EDO only, other EDOs have other correlations. | You can loosely relate the ups and downs to JI: major = red or fifthward white, downmajor = yellow, upminor = green, minor = blue or fourthwards white. Or simply up = green, down = yellow, and mid = white, blue or red. (See [[Kite's color notation]] for an explanation of the colors.) These correlations are for 22-EDO only, other EDOs have other correlations. | ||
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pentatonic EDOs, with a fifth = 720¢ | pentatonic EDOs, with a fifth = 720¢ | ||
"sweet" EDOs, so-called because the fifth hits the "sweet spot" between 720¢ and 686¢ | "sweet" EDOs, so-called because the fifth hits the "sweet spot" between 720¢ and 686¢ | ||
heptatonic EDOs, with a fifth = four sevenths of an octave = 686¢ | heptatonic EDOs, with a fifth = four sevenths of an octave = 4\7 = 686¢ | ||
superflat EDOs or Mavila EDOs, with a fifth less than 686¢ | superflat EDOs or Mavila EDOs, with a fifth less than 686¢ | ||
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As we've seen, 19-EDO doesn't require ups and downs. Let the keyspan of the octave in an EDO be K1 and the keyspan of the fifth be K2. For example, in 12-EDO, K1 = 12 and K2 = 7. The stepspan is one less than the degree. For our usual heptatonic framework, the stepspan of the octave S1 is 7 and the stepspan of the fifth S2 is 4. In order for ups and downs to be unnecessary, S1 * K2 - S2 * K1 = +/-1. Examples of EDOs that don't need ups and downs are 5, 12, 19, 26, 33, 40, etc. (every 7th EDO). There are 4 other such EDOs, 7, 9, 16 and 23. All other EDOs need ups and downs. | As we've seen, 19-EDO doesn't require ups and downs. Let the keyspan of the octave in an EDO be K1 and the keyspan of the fifth be K2. For example, in 12-EDO, K1 = 12 and K2 = 7. The stepspan is one less than the degree. For our usual heptatonic framework, the stepspan of the octave S1 is 7 and the stepspan of the fifth S2 is 4. In order for ups and downs to be unnecessary, S1 * K2 - S2 * K1 = +/-1. Examples of EDOs that don't need ups and downs are 5, 12, 19, 26, 33, 40, etc. (every 7th EDO). There are 4 other such EDOs, 7, 9, 16 and 23. All other EDOs need ups and downs. | ||
**__17-EDO__:** | **__17-EDO__:** (2 keys per sharp/flat) | ||
Black and white keys: C | Black and white keys: C * * D * * E F * * G * * A * * B C | ||
Relative notation: P1 m2 vM2 M2 m3 vM3 M3 P4 d5 vP5 P5 m6 vM6 M6 m7 vM7 M7 P8 | Relative notation: P1 m2 vM2 M2 m3 vM3 M3 P4 d5 vP5 P5 m6 vM6 M6 m7 vM7 M7 P8 | ||
or with upminors instead of downmajors: P1 m2 ^m2 M2 m3 ^m3 M3 P4 d5 ^d5 P5 m6 ^m6 M6 m7 ^m7 M7 P8 | or with upminors instead of downmajors: P1 m2 ^m2 M2 m3 ^m3 M3 P4 d5 ^d5 P5 m6 ^m6 M6 m7 ^m7 M7 P8 | ||
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In C, with downmajors: C Db Dv D Eb Ev E F Gb Gv G Ab Av A Bb Bv B C | In C, with downmajors: C Db Dv D Eb Ev E F Gb Gv G Ab Av A Bb Bv B C | ||
In B, with upminors: B C C^ C# D D^ D# E F F^ F# G G^ G# A A^ A# B | In B, with upminors: B C C^ C# D D^ D# E F F^ F# G G^ G# A A^ A# B | ||
One can't associate ups and downs with yellow and green because of the poor approximation of the 5-limit. However major = red or fifthward white, minor = blue or fourthward white, and downmajor = upminor = jade or amber. | |||
**__24-EDO__:** (2 keys per sharp/flat) | |||
black and white keys: C * * * D * * * E * F * * * G * * * A * * * B * C | |||
**__24-EDO__:** | |||
black and white keys: C | |||
Relative notation: P1 vm2 m2 vM2 M2 vm3 m3 vM3 M3 vP4 P4 ^P4 d5 vP5 P5 etc. | Relative notation: P1 vm2 m2 vM2 M2 vm3 m3 vM3 M3 vP4 P4 ^P4 d5 vP5 P5 etc. | ||
Many alternate spellings available, for example vm3 = ^M2, vM3 = ^m3, ^P4 = vd5, etc. | Many alternate spellings available, for example vm3 = ^M2, vM3 = ^m3, ^P4 = vd5, etc. | ||
In C: C Dbv Db Dv D Ebv Eb Ev E Fv F F^ Gb Gv G etc. | In C: C Dbv Db Dv D Ebv Eb Ev E Fv F F^ Gb Gv G etc. | ||
JI associations: Major = yellow or fifthward white, minor = green or fourthward white, upmajor = red, downminor = blue, downmajor = upminor = jade or amber. | |||
24-EDO is an example of a closed EDO. An EDO is closed if the keyspan of the fifth isn't coprime with the keyspan of the octave, and open if it is. 24-EDO has a fifth of 14 steps, and 14 isn't coprime with 24, because they have a common divisor of 2. 24-EDO is said to close at 12 (1/2 of 24), because the circle of fifths has only 12 notes. There are actually 2 unconnected circles of fifths in 24-EDO, which are notated as the mid one and the up one: | 24-EDO is an example of a closed EDO. An EDO is closed if the keyspan of the fifth isn't coprime with the keyspan of the octave, and open if it is. 24-EDO has a fifth of 14 steps, and 14 isn't coprime with 24, because they have a common divisor of 2. 24-EDO is said to close at 12 (1/2 of 24), because the circle of fifths has only 12 notes. There are actually 2 unconnected circles of fifths in 24-EDO, which are notated as the mid one and the up one: | ||
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C^ Db Db^ D D^ Eb Eb^ E E^ F F^ F^^ Gb^ G G^ etc. | C^ Db Db^ D D^ Eb Eb^ E E^ F F^ F^^ Gb^ G G^ etc. | ||
**__31-EDO__:** (2 keys per sharp/flat) | |||
**__31-EDO__:** | |||
Black and white keys: C * * * * D * * * * E * * F * * * * G * * * * A * * * * B * * C | Black and white keys: C * * * * D * * * * E * * F * * * * G * * * * A * * * * B * * C | ||
relative notation: P1 ^P1 vm2 m2 ^m2 M2 ^M2 vm3 m3 ^m3 M3 ^M3 vP4 P4 ^P4 A4 d5 ^d5 P5 etc. | relative notation: P1 ^P1 vm2 m2 ^m2 M2 ^M2 vm3 m3 ^m3 M3 ^M3 vP4 P4 ^P4 A4 d5 ^d5 P5 etc. | ||
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In C: C C^ Dbv Db Db^ D D^ Ebv Eb Eb^ E E^ Fv F F^ F# Gb Gb^ G etc. | In C: C C^ Dbv Db Db^ D D^ Ebv Eb Eb^ E E^ Fv F F^ F# Gb Gb^ G etc. | ||
JI associations: Perfect = white, major = yellow or fifthward white, minor = green or fourthward white, downminor = blue, upmajor = red, downmajor = upminor = jade or amber (same as 24-EDO). | JI associations: Perfect = white, major = yellow or fifthward white, minor = green or fourthward white, downminor = blue, upmajor = red, downmajor = upminor = jade or amber (same as 24-EDO). | ||
**__41-EDO__:** (4 keys per sharp/flat) | |||
Black and white keys: C * * * * * * D * * * * * * E * * F * * * * * * G * * * * * * A * * * * * * B * * C | |||
P1 ^P1 vm2 m2 ^m2 ^^m2 vM2 M2 ^M2 vm3 m3 ^m3 ^^m3 vM3 M3 ^M3 vP4 P4 ^P4 ^^P4 d5 ^d5 vvP5 vP5 P5 etc. | |||
alternate spellings: A1=^m2, ^^m2=vvM2, ^M3=vP4, vA4=d5, A4=^d5, etc. | |||
In C: C C^ Dbv Db Db^ D D^ Ebv Eb Eb^ E E^ Fv F F^ F# Gb Gb^ G etc. | |||
JI associations: Perfect = white, major = fifthward white, minor = fourthward white, downmajor = yellow, upminor = green, downminor = blue, upmajor = red, double-downmajor = double-upminor = jade or amber. | |||
=__Naming Chords__= | =__Naming Chords__= | ||
Ups and downs allow us to name any chord easily. First we need an exact definition of major, minor, perfect, etc. that works with all edos. The quality of an interval is defined by its position on the chain of 5ths. Perfect is 0-1 steps away, major/minor are 2-5 steps away, aug/dim are 6-12 steps away, etc. | Ups and downs allow us to name any chord easily. First we need an exact definition of major, minor, perfect, etc. that works with all edos. The quality of an interval is defined by its position on the chain of 5ths (or more generally, the chain of generators). Perfect is 0-1 steps away, major/minor are 2-5 steps away, aug/dim are 6-12 steps away, etc. | ||
There are 3 special cases to be addressed. The first is when the edo's 5th is narrower than 4\7, as in 16edo. Major is defined as wider than minor, so major is not fifthwards but fourthwards: | There are 3 special cases to be addressed. The first is when the edo's 5th is narrower than 4\7, as in 16edo. Major is defined as wider than minor, so major is not fifthwards but fourthwards: | ||
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Eb + m3 --> E# + M3 = G## --> Gbb | Eb + m3 --> E# + M3 = G## --> Gbb | ||
The second special case is when the edo's fifth equals 4\7, as in 7edo, 14edo, 21edo, 28edo, and 35edo. (42edo, 49edo, etc. have a fifth wider than 4\7.) In these five edos, there are zero keys per sharp/flat, and all intervals are perfect. | The second special case is when the edo's fifth equals 4\7, as in 7edo, 14edo, 21edo, 28edo, and 35edo. (42edo, 49edo, etc. have a fifth wider than 4\7.) In these five edos, there are zero keys per sharp/flat, and all intervals are perfect. That's because the scale that is produced by a chain of fifths is exactly the same scale as produced by a chain of 2nds, 3rds, 4ths, etc. Since any of these intervals is a potential generator, and since the generator is perfect by definition, they must all be perfect. | ||
The chain of fifths in heptatonic EDOs (3/2 maps to 4\7): | The chain of fifths in heptatonic EDOs (3/2 maps to 4\7): | ||
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"Fifth-less" EDOs (8, 11, 13 and 18) | "Fifth-less" EDOs (8, 11, 13 and 18) | ||
Fourthward EDOs (9, 16 and 23) | |||
Heptatonic EDOs (7, 14, 21, 28 and 35) | |||
Pentatonic EDOs (5, 10, 15, 20, 25 and 30) | |||
All others | |||
**__8edo__:** (generator = 1\8 2nd) D E F G * A B C D | ===__**"Fifth-less" EDOs (8, 11, 13 and 18)**__=== | ||
**__8edo__:** (generator = 1\8 2nd) | |||
D E F G * A B C D | |||
P1 - P2 - m3 - M3/m4 - M4/m5 - M5/m6 - M6 - P7 - P8 | P1 - P2 - m3 - M3/m4 - M4/m5 - M5/m6 - M6 - P7 - P8 | ||
chain of | seventhwards chain of seconds: M3 - M4 - M5 - M6 - P7 - P1 - P2 - m3 - m4 - m5 - m6 - d7 etc. | ||
A# - B# - C# - D# - E# - F# - G# - A - B - C - D - E - F - G - Ab - Bb - Cb - Db - Eb - Fb - Gb</pre></div> | A# - B# - C# - D# - E# - F# - G# - A - B - C - D - E - F - G - Ab - Bb - Cb - Db - Eb - Fb - Gb | ||
__**11edo**__: (generator = 3\11 3rd) | |||
D * E F * G A * B C * D | |||
P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8 | |||
sixthwards chain of thirds: M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 etc. | |||
E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb | |||
__**13edo**__**:** (generator = 2\13 2nd) | |||
D * E * F * G A * B * C * D | |||
P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - m6 - M6 - P7 - A7/d8 - P8 | |||
secondwards chain of seconds: m3 - m4 - m5 - m6 - P7 - P1 - P2 - M3 - M4 - M5 - M6 - A7 etc. | |||
Ab - Bb - Cb - Db - Eb - Fb - Gb - A - B - C - D - E - F - G - A# - B# - C# - D# - E# - F# - G# | |||
**__18edo__:** (generator = 5\18 = 3rd) | |||
D * * E * F * * G * A * * B * C * * D | |||
P1 - A1/d2 - m2 - M2 - A2/d3 - P3 - A3/d4 - m4 - M4 - A4/d5 - m5 - M5 - A5/d6 - P6 - A6/d7 - m7 - M7 - A7/d8 - P8 | |||
sixthwards chain of thirds: M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 etc. | |||
E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb | |||
===__Fourthward EDOs (9, 16 and 23)__=== </pre></div> | |||
<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Ups and Downs Notation</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x&quot;Ups and Downs&quot; Notation"></a><!-- ws:end:WikiTextHeadingRule:0 -->&quot;Ups and Downs&quot; Notation</h1> | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Ups and Downs Notation</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x&quot;Ups and Downs&quot; Notation"></a><!-- ws:end:WikiTextHeadingRule:0 -->&quot;Ups and Downs&quot; Notation</h1> | ||
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The advantage to this notation is that you always know where your fifth is. And hence your 4th, and your major 9th, hence the maj 2nd and the min 7th too. You have convenient landmarks to find your way around, built into the notation. The notation is a map of unfamiliar territory, and we want this map to be as easy to read as possible.<br /> | The advantage to this notation is that you always know where your fifth is. And hence your 4th, and your major 9th, hence the maj 2nd and the min 7th too. You have convenient landmarks to find your way around, built into the notation. The notation is a map of unfamiliar territory, and we want this map to be as easy to read as possible.<br /> | ||
<br /> | <br /> | ||
The basic pattern for 22-EDO is P1-m2-^m2-vM2-M2-m3-^m3-vM3-M3-P4-d5-^d5-vP5-P5 etc. That's pronounced &quot;upminor 2nd, downmajor 3rd&quot;, etc. The ups and downs are leading in relative notation but trailing in absolute notation. You can apply this pattern to any key, with certain keys requiring double-sharps or even triple-sharps. The mid notes always form a | The basic pattern for 22-EDO is P1-m2-^m2-vM2-M2-m3-^m3-vM3-M3-P4-d5-^d5-vP5-P5 etc. That's pronounced &quot;upminor 2nd, downmajor 3rd&quot;, etc. The ups and downs are leading in relative notation but trailing in absolute notation. You can apply this pattern to any key, with certain keys requiring double-sharps or even triple-sharps. The mid notes always form a chain of fifths.<br /> | ||
<br /> | <br /> | ||
You can loosely relate the ups and downs to JI: major = red or fifthward white, downmajor = yellow, upminor = green, minor = blue or fourthwards white. Or simply up = green, down = yellow, and mid = white, blue or red. (See <a class="wiki_link" href="/Kite%27s%20color%20notation">Kite's color notation</a> for an explanation of the colors.) These correlations are for 22-EDO only, other EDOs have other correlations.<br /> | You can loosely relate the ups and downs to JI: major = red or fifthward white, downmajor = yellow, upminor = green, minor = blue or fourthwards white. Or simply up = green, down = yellow, and mid = white, blue or red. (See <a class="wiki_link" href="/Kite%27s%20color%20notation">Kite's color notation</a> for an explanation of the colors.) These correlations are for 22-EDO only, other EDOs have other correlations.<br /> | ||
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pentatonic EDOs, with a fifth = 720¢<br /> | pentatonic EDOs, with a fifth = 720¢<br /> | ||
&quot;sweet&quot; EDOs, so-called because the fifth hits the &quot;sweet spot&quot; between 720¢ and 686¢<br /> | &quot;sweet&quot; EDOs, so-called because the fifth hits the &quot;sweet spot&quot; between 720¢ and 686¢<br /> | ||
heptatonic EDOs, with a fifth = four sevenths of an octave = 686¢<br /> | heptatonic EDOs, with a fifth = four sevenths of an octave = 4\7 = 686¢<br /> | ||
superflat EDOs or Mavila EDOs, with a fifth less than 686¢<br /> | superflat EDOs or Mavila EDOs, with a fifth less than 686¢<br /> | ||
<br /> | <br /> | ||
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As we've seen, 19-EDO doesn't require ups and downs. Let the keyspan of the octave in an EDO be K1 and the keyspan of the fifth be K2. For example, in 12-EDO, K1 = 12 and K2 = 7. The stepspan is one less than the degree. For our usual heptatonic framework, the stepspan of the octave S1 is 7 and the stepspan of the fifth S2 is 4. In order for ups and downs to be unnecessary, S1 * K2 - S2 * K1 = +/-1. Examples of EDOs that don't need ups and downs are 5, 12, 19, 26, 33, 40, etc. (every 7th EDO). There are 4 other such EDOs, 7, 9, 16 and 23. All other EDOs need ups and downs.<br /> | As we've seen, 19-EDO doesn't require ups and downs. Let the keyspan of the octave in an EDO be K1 and the keyspan of the fifth be K2. For example, in 12-EDO, K1 = 12 and K2 = 7. The stepspan is one less than the degree. For our usual heptatonic framework, the stepspan of the octave S1 is 7 and the stepspan of the fifth S2 is 4. In order for ups and downs to be unnecessary, S1 * K2 - S2 * K1 = +/-1. Examples of EDOs that don't need ups and downs are 5, 12, 19, 26, 33, 40, etc. (every 7th EDO). There are 4 other such EDOs, 7, 9, 16 and 23. All other EDOs need ups and downs.<br /> | ||
<br /> | <br /> | ||
<strong><u>17-EDO</u>:</strong><br /> | <strong><u>17-EDO</u>:</strong> (2 keys per sharp/flat)<br /> | ||
Black and white keys: C | Black and white keys: C * * D * * E F * * G * * A * * B C<br /> | ||
Relative notation: P1 m2 vM2 M2 m3 vM3 M3 P4 d5 vP5 P5 m6 vM6 M6 m7 vM7 M7 P8<br /> | Relative notation: P1 m2 vM2 M2 m3 vM3 M3 P4 d5 vP5 P5 m6 vM6 M6 m7 vM7 M7 P8<br /> | ||
or with upminors instead of downmajors: P1 m2 ^m2 M2 m3 ^m3 M3 P4 d5 ^d5 P5 m6 ^m6 M6 m7 ^m7 M7 P8<br /> | or with upminors instead of downmajors: P1 m2 ^m2 M2 m3 ^m3 M3 P4 d5 ^d5 P5 m6 ^m6 M6 m7 ^m7 M7 P8<br /> | ||
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In C, with downmajors: C Db Dv D Eb Ev E F Gb Gv G Ab Av A Bb Bv B C<br /> | In C, with downmajors: C Db Dv D Eb Ev E F Gb Gv G Ab Av A Bb Bv B C<br /> | ||
In B, with upminors: B C C^ C# D D^ D# E F F^ F# G G^ G# A A^ A# B<br /> | In B, with upminors: B C C^ C# D D^ D# E F F^ F# G G^ G# A A^ A# B<br /> | ||
One can't associate ups and downs with yellow and green because of the poor approximation of the 5-limit. However major = red or fifthward white, minor = blue or fourthward white, and downmajor = upminor = jade or amber.<br /> | |||
<br /> | <br /> | ||
<strong><u>24-EDO</u>:</strong> (2 keys per sharp/flat)<br /> | |||
black and white keys: C * * * D * * * E * F * * * G * * * A * * * B * C<br /> | |||
<strong><u>24-EDO</u>:</strong><br /> | |||
black and white keys: C | |||
Relative notation: P1 vm2 m2 vM2 M2 vm3 m3 vM3 M3 vP4 P4 ^P4 d5 vP5 P5 etc.<br /> | Relative notation: P1 vm2 m2 vM2 M2 vm3 m3 vM3 M3 vP4 P4 ^P4 d5 vP5 P5 etc.<br /> | ||
Many alternate spellings available, for example vm3 = ^M2, vM3 = ^m3, ^P4 = vd5, etc.<br /> | Many alternate spellings available, for example vm3 = ^M2, vM3 = ^m3, ^P4 = vd5, etc.<br /> | ||
In C: C Dbv Db Dv D Ebv Eb Ev E Fv F F^ Gb Gv G etc.<br /> | In C: C Dbv Db Dv D Ebv Eb Ev E Fv F F^ Gb Gv G etc.<br /> | ||
JI associations: Major = yellow or fifthward white, minor = green or fourthward white, upmajor = red, downminor = blue, downmajor = upminor = jade or amber.<br /> | |||
<br /> | <br /> | ||
24-EDO is an example of a closed EDO. An EDO is closed if the keyspan of the fifth isn't coprime with the keyspan of the octave, and open if it is. 24-EDO has a fifth of 14 steps, and 14 isn't coprime with 24, because they have a common divisor of 2. 24-EDO is said to close at 12 (1/2 of 24), because the circle of fifths has only 12 notes. There are actually 2 unconnected circles of fifths in 24-EDO, which are notated as the mid one and the up one:<br /> | 24-EDO is an example of a closed EDO. An EDO is closed if the keyspan of the fifth isn't coprime with the keyspan of the octave, and open if it is. 24-EDO has a fifth of 14 steps, and 14 isn't coprime with 24, because they have a common divisor of 2. 24-EDO is said to close at 12 (1/2 of 24), because the circle of fifths has only 12 notes. There are actually 2 unconnected circles of fifths in 24-EDO, which are notated as the mid one and the up one:<br /> | ||
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C^ Db Db^ D D^ Eb Eb^ E E^ F F^ F^^ Gb^ G G^ etc.<br /> | C^ Db Db^ D D^ Eb Eb^ E E^ F F^ F^^ Gb^ G G^ etc.<br /> | ||
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<strong><u>31-EDO</u>:</strong> (2 keys per sharp/flat)<br /> | |||
<strong><u>31-EDO</u>:</strong><br /> | |||
Black and white keys: C * * * * D * * * * E * * F * * * * G * * * * A * * * * B * * C<br /> | Black and white keys: C * * * * D * * * * E * * F * * * * G * * * * A * * * * B * * C<br /> | ||
relative notation: P1 ^P1 vm2 m2 ^m2 M2 ^M2 vm3 m3 ^m3 M3 ^M3 vP4 P4 ^P4 A4 d5 ^d5 P5 etc.<br /> | relative notation: P1 ^P1 vm2 m2 ^m2 M2 ^M2 vm3 m3 ^m3 M3 ^M3 vP4 P4 ^P4 A4 d5 ^d5 P5 etc.<br /> | ||
| Line 574: | Line 605: | ||
In C: C C^ Dbv Db Db^ D D^ Ebv Eb Eb^ E E^ Fv F F^ F# Gb Gb^ G etc.<br /> | In C: C C^ Dbv Db Db^ D D^ Ebv Eb Eb^ E E^ Fv F F^ F# Gb Gb^ G etc.<br /> | ||
JI associations: Perfect = white, major = yellow or fifthward white, minor = green or fourthward white, downminor = blue, upmajor = red, downmajor = upminor = jade or amber (same as 24-EDO).<br /> | JI associations: Perfect = white, major = yellow or fifthward white, minor = green or fourthward white, downminor = blue, upmajor = red, downmajor = upminor = jade or amber (same as 24-EDO).<br /> | ||
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<strong><u>41-EDO</u>:</strong> (4 keys per sharp/flat)<br /> | |||
Black and white keys: C * * * * * * D * * * * * * E * * F * * * * * * G * * * * * * A * * * * * * B * * C<br /> | |||
P1 ^P1 vm2 m2 ^m2 ^^m2 vM2 M2 ^M2 vm3 m3 ^m3 ^^m3 vM3 M3 ^M3 vP4 P4 ^P4 ^^P4 d5 ^d5 vvP5 vP5 P5 etc.<br /> | |||
alternate spellings: A1=^m2, ^^m2=vvM2, ^M3=vP4, vA4=d5, A4=^d5, etc.<br /> | |||
In C: C C^ Dbv Db Db^ D D^ Ebv Eb Eb^ E E^ Fv F F^ F# Gb Gb^ G etc.<br /> | |||
JI associations: Perfect = white, major = fifthward white, minor = fourthward white, downmajor = yellow, upminor = green, downminor = blue, upmajor = red, double-downmajor = double-upminor = jade or amber.<br /> | |||
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<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Naming Chords"></a><!-- ws:end:WikiTextHeadingRule:2 --><u>Naming Chords</u></h1> | <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Naming Chords"></a><!-- ws:end:WikiTextHeadingRule:2 --><u>Naming Chords</u></h1> | ||
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Ups and downs allow us to name any chord easily. First we need an exact definition of major, minor, perfect, etc. that works with all edos. The quality of an interval is defined by its position on the chain of 5ths. Perfect is 0-1 steps away, major/minor are 2-5 steps away, aug/dim are 6-12 steps away, etc.<br /> | Ups and downs allow us to name any chord easily. First we need an exact definition of major, minor, perfect, etc. that works with all edos. The quality of an interval is defined by its position on the chain of 5ths (or more generally, the chain of generators). Perfect is 0-1 steps away, major/minor are 2-5 steps away, aug/dim are 6-12 steps away, etc.<br /> | ||
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There are 3 special cases to be addressed. The first is when the edo's 5th is narrower than 4\7, as in 16edo. Major is defined as wider than minor, so major is not fifthwards but fourthwards:<br /> | There are 3 special cases to be addressed. The first is when the edo's 5th is narrower than 4\7, as in 16edo. Major is defined as wider than minor, so major is not fifthwards but fourthwards:<br /> | ||
| Line 592: | Line 631: | ||
Eb + m3 --&gt; E# + M3 = G## --&gt; Gbb<br /> | Eb + m3 --&gt; E# + M3 = G## --&gt; Gbb<br /> | ||
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The second special case is when the edo's fifth equals 4\7, as in 7edo, 14edo, 21edo, 28edo, and 35edo. (42edo, 49edo, etc. have a fifth wider than 4\7.) In these five edos, there are zero keys per sharp/flat, and all intervals are perfect.<br /> | The second special case is when the edo's fifth equals 4\7, as in 7edo, 14edo, 21edo, 28edo, and 35edo. (42edo, 49edo, etc. have a fifth wider than 4\7.) In these five edos, there are zero keys per sharp/flat, and all intervals are perfect. That's because the scale that is produced by a chain of fifths is exactly the same scale as produced by a chain of 2nds, 3rds, 4ths, etc. Since any of these intervals is a potential generator, and since the generator is perfect by definition, they must all be perfect.<br /> | ||
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The chain of fifths in heptatonic EDOs (3/2 maps to 4\7):<br /> | The chain of fifths in heptatonic EDOs (3/2 maps to 4\7):<br /> | ||
| Line 792: | Line 831: | ||
Not counting the trivial edos 2, 3, 4 and 6, there are only seven such edos. As seen in this diagram, they are the ones to the left of the central line in the light blue region, plus the ones to the right of the central line in the orange region. The ones on the left edge of the blue region are the fourthward ones like 16edo, and have been dealt with already. 23edo can be notated similarly to 16edo by using a fifth of 13\23 instead of 14\23. That leaves only four edos: 8, 11, 13, and 18.<br /> | Not counting the trivial edos 2, 3, 4 and 6, there are only seven such edos. As seen in this diagram, they are the ones to the left of the central line in the light blue region, plus the ones to the right of the central line in the orange region. The ones on the left edge of the blue region are the fourthward ones like 16edo, and have been dealt with already. 23edo can be notated similarly to 16edo by using a fifth of 13\23 instead of 14\23. That leaves only four edos: 8, 11, 13, and 18.<br /> | ||
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&quot;Fifth-less&quot; EDOs (8, 11, 13 and 18)<br /> | &quot;Fifth-less&quot; EDOs (8, 11, 13 and 18)<br /> | ||
Fourthward EDOs (9, 16 and 23)<br /> | |||
Heptatonic EDOs (7, 14, 21, 28 and 35)<br /> | |||
Pentatonic EDOs (5, 10, 15, 20, 25 and 30)<br /> | |||
All others<br /> | |||
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<strong><u>8edo</u>:</strong> (generator = 1\8 2nd) D E F G * A B C D<br /> | <!-- ws:start:WikiTextHeadingRule:14:&lt;h3&gt; --><h3 id="toc7"><a name="Summary of EDO notation--&quot;Fifth-less&quot; EDOs (8, 11, 13 and 18)"></a><!-- ws:end:WikiTextHeadingRule:14 --><u><strong>&quot;Fifth-less&quot; EDOs (8, 11, 13 and 18)</strong></u></h3> | ||
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<strong><u>8edo</u>:</strong> (generator = 1\8 2nd) <br /> | |||
D E F G * A B C D<br /> | |||
P1 - P2 - m3 - M3/m4 - M4/m5 - M5/m6 - M6 - P7 - P8<br /> | P1 - P2 - m3 - M3/m4 - M4/m5 - M5/m6 - M6 - P7 - P8<br /> | ||
chain of | seventhwards chain of seconds: M3 - M4 - M5 - M6 - P7 - P1 - P2 - m3 - m4 - m5 - m6 - d7 etc.<br /> | ||
A# - B# - C# - D# - E# - F# - G# - A - B - C - D - E - F - G - Ab - Bb - Cb - Db - Eb - Fb - Gb</body></html></pre></div> | A# - B# - C# - D# - E# - F# - G# - A - B - C - D - E - F - G - Ab - Bb - Cb - Db - Eb - Fb - Gb<br /> | ||
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<u><strong>11edo</strong></u>: (generator = 3\11 3rd) <br /> | |||
D * E F * G A * B C * D <br /> | |||
P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8<br /> | |||
sixthwards chain of thirds: M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 etc.<br /> | |||
E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb<br /> | |||
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<u><strong>13edo</strong></u><strong>:</strong> (generator = 2\13 2nd)<br /> | |||
D * E * F * G A * B * C * D<br /> | |||
P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - m6 - M6 - P7 - A7/d8 - P8<br /> | |||
secondwards chain of seconds: m3 - m4 - m5 - m6 - P7 - P1 - P2 - M3 - M4 - M5 - M6 - A7 etc.<br /> | |||
Ab - Bb - Cb - Db - Eb - Fb - Gb - A - B - C - D - E - F - G - A# - B# - C# - D# - E# - F# - G#<br /> | |||
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<strong><u>18edo</u>:</strong> (generator = 5\18 = 3rd)<br /> | |||
D * * E * F * * G * A * * B * C * * D<br /> | |||
P1 - A1/d2 - m2 - M2 - A2/d3 - P3 - A3/d4 - m4 - M4 - A4/d5 - m5 - M5 - A5/d6 - P6 - A6/d7 - m7 - M7 - A7/d8 - P8<br /> | |||
sixthwards chain of thirds: M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 etc.<br /> | |||
E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb<br /> | |||
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<!-- ws:start:WikiTextHeadingRule:16:&lt;h3&gt; --><h3 id="toc8"><a name="Summary of EDO notation--Fourthward EDOs (9, 16 and 23)"></a><!-- ws:end:WikiTextHeadingRule:16 --><u>Fourthward EDOs (9, 16 and 23)</u></h3> | |||
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